Plasma Jets and Stellar Lightnings Involved in Coronal Heating Dynamics

1. Introduction

A star’s luminosity (its total power output) is determined by its surface temperature and its radius (size), using the Stefan-Boltzmann Law. The luminosity of the Sun is used as the standard unit of measurement (1 solar luminosity ) for comparing with the luminosity of other stars. The surface temperature of the Sun is approximately 5,000 Kelvin (ranging between 5,772 K to 6000 K across different models). The surface temperature and Lumosity of a star are related to the mass, radius, and amount of fuel that is burning in the core of a star. The low-mass star can burn hydrogen to form helium gas. The massive stars may burn multiple chemical elements at once to form several elements and release a sufficient amount of energy into free space. The “Coronal Heating Problem” persists because multiple mechanisms likely operate simultaneously as Magnetic Reconnection (DC Heating), Wave Dissipation (AC Heating), Magnetohydrodynamic (MHD) waves as the Alfvén waves, and Current Sheet Dissipation. Powerful plasma jets and rapid lightning trees come out from the interiors of the Sun and stars and are involved in solar and stellar coronal heating dynamics. The heating dynamics of stellar coronae are the rapid increase in the outer shell of stars and the Sun to millions of degrees. It is a dynamic process driven by the equilibrium between magnetic pressure and thermal pressure. In this environment, magnetic pressure dominates, confining the plasma in loops, while thermal pressure tends to expand it. The continuous struggle between these two forces dictates how magnetic energy is released, acting as a “self-regulating cycle” that keeps the corona hot. Hottest and energetic plasma particles rising from an interior shell of a star to enhance the coronal heating dynamics and release energetic particles freely into space with a speed higher than the escape velocity from the external surface of stars [1,2]. Solar and stellar activities are included: X-ray emission from stellar coronae, enhanced chromosphere emission during stellar activity, is notably prominent in the (Hydrogen-alpha) line and the Ca II H & K resonance lines, and Visible and radio emission from stellar flares [3–7]. The central temperature and pressure in the center of stars increased violently to enhance the chains of nuclear fusion processes and release powerful plasma jets and stellar lightnings into space. Indeed, Powerful plasma jets and stellar lightning are coming from massive stars; typically, these stars have sufficient central pressure and temperature to produce heavy elements such as silicon, germanium, copper, manganese, and magnesium. Plasma jets and multiple stellar lightings that are happening at once, such as lightning trees, are coming out from interior shell of massive stars as evidence of violent events that happening in the core of stars and their external surface. The jet speed of plasma and stellar lighting may exceed the escape velocity of the stars. As a result, high-energy particles may come out directly from the outer surface of stars and the Sun. high energy particles are coming out from the Sun and stars rapidly with higher speed and momentum. The plasma jets and stellar lightnings participate directly in the turbulence, magnetic hydrodynamic waves, and heating up of the stellar atmosphere and coronal heating dynamics. The figures and images in this research work are based on theoretical concepts and sketches obtained by using the advanced MATLAB program and mathematical equations to describe coronal heating dynamics clearly.

The main objectives of this study are to calculate the escape velocity of energetic particles v, the Radius R, Surface Temperature T, and luminosity L of the stars.

2. An Escape Velocity and Temperature of the Energetic Particles in Plasma Jets

An escape velocity of the energetic particles at a distance r from interior and surface of the massive stars can be described by a Newtonian mechanic and thermodynamics. The wave lengthen, frequency, mass, and typically the temperature of accelerated particles had been used to calculate the escape velocity and kinetic energy of particles. The temperature of accelerated particles and ions that come out from depth of stars are depending on the interior temperature and pressure of stars. Following the general equations of the Gravitational acceleration$\mathit{ }\mathit{g}\left(\mathit{r}\right)$ and Potential energy $\mathit{\varphi }\left(\mathit{r}\right)$ of the plasma jets and stellar lightnings at a distance $\mathit{r}$ from the external surface of the massive stars:

$\mathit{g}\left(\mathit{r}\right)={\displaystyle \frac{\mathit{G}\mathit{M}}{{\mathit{r}}^2}}$  (1)

$\mathit{\varphi }\left(\mathit{r}\right)=\mathit{ }{\displaystyle \frac{\mathit{G}\mathit{M}}{\mathit{r}}}$  (2)

Following is Wien’s Law, which plays a crucial role in measuring the surface temperature of the hottest stars and wavelength of the radiation which radiated from celestial objects:

${\mathit{\lambda }}_{\mathit{M}\mathit{a}\mathit{x}}={\displaystyle \frac{0.0029\mathit{ }\mathit{m}.\mathit{ }\mathit{K}}{\mathit{T}}}$  (3)

where E is a thermal energy of particles and celestial objects, and ${\mathit{\lambda }}_{\mathit{M}\mathit{a}\mathit{x}}$ is the maximum wavelength of a photon particle, it is inversely proportional with a temperature T. If potential energy $\mathit{\varphi }\left(\mathit{r}\right)$ and Kinetic Energy $(KE={\displaystyle \frac{1}{2}}\mathit{m}{\mathit{v}}^2)$ were equalized, yield to:

$\mathit{v}=\sqrt{{\displaystyle \frac{2\mathit{G}\mathit{ }\mathit{M}\mathbf{ }}{\mathit{r}}}}$  (4)

where $\left(\mathit{M}\mathit{ }\right)$ is the Mass of Star, $\mathit{r}$ is the radius of a star, and $\mathit{v}$ is the escape velocity of plasma jets , ions, and energetic particles from the external surface of a star at the speed lower than the speed of light in space. Thusly, a gravitational constant is $( G=6.674 x {10}^{-11}\mathrm{ }{\displaystyle \frac{N.m^2}{{kg}^2}}$), and$\mathit{ }\mathit{m}$ is the mass of accelerated particles which escaped at a distance r from massive stars. If kinetic energy of particles in the hottest plasma jets and lightning trees $(KE={\displaystyle \frac{1}{2}}\mathit{m}{\mathit{v}}^2)$ were thermalized $(E= {\displaystyle \frac{3}{2}}\mathit{k}\mathbf{T})$ yield to:

${\displaystyle \frac{1}{2}}\mathit{m}{\mathit{v}}^2={\displaystyle \frac{3}{2}}\mathit{k}\mathbf{T}$  (5)

$\mathit{T}={\displaystyle \frac{\mathit{m}{\mathit{v}}^2}{3{\mathit{K}}_{\mathit{B}}}}$  (6)

where $\mathit{T}$ is the temperature of accelerated particles and plasma jets from external surface of the stars was changed immediately due to the gravity, electromagnetic field, rapid temperature change, nuclear fusion dynamic, central temperature effects, and internal thermal pressure of stars. The Boltzmann constant $\left({\mathit{K}}_{\mathit{B}}=1.38\mathit{ }\mathit{x}\mathit{ }{10}^{-23}\mathit{ }\mathit{J}/\mathit{K}\mathit{ }\right)$ plays a crucial role, and utilized in this mechanism. Escape velocity is the minimum speed an object requires to break free from a celestial body’s gravitational pull without further propulsion. On a black hole, it is 300,000 km/s; on Earth, this threshold is approximately 11.2 km/s. To escape a gravitational field, an object’s total mechanical energy must be at least zero at the surface, since the sum of kinetic energy and gravitational potential energy is zero. An escape velocity on the surface of stars is lower than the speed of light. But on the surface of a black hole singularity, an escape velocity may exceed the speed of light [8–14]. The singularity has a tiny size as an atom, and only superparticles can escape from its powerful gravity. An object with a mass of stars, planets, and galaxies had been compressed into the size of atoms and incredible density. The speed of superparticles can exceed the speed of light to collide with a black hole singularity or leave it forever. The theory of escape velocity was developed from Newtonian mechanics, General relativity, and quantum mechanics, and applied to escape velocity from the surface of a white, smooth ball of a singularity. The singularity is a compact, physical object with a tiny size, immense gravity, maximum surface temperature, huge thermal energy, central pressure, and high angular momentum.

Problem1: Calculate an escape velocity v and T temperature of particles as powerful plasma jets come out from surface of a blue star with a Mass 20 times the mass of Sun and its radius 8 times the radius of a Sun while escaping at speed 100 kilometers per second to 1000 kilometers per second ?.

The mass of a blue massive star is approximately 20 times of the solar masses $\left({\mathit{M}=20\mathit{ }\mathit{x}\mathit{ }\mathit{M}}_{\mathit{S}\mathit{u}\mathit{n}}=3.98\mathit{ }\mathit{x}{10}^{31}\mathit{k}\mathit{g}\right)$, and its radius could be eight times the radius of a Sun ${(\mathit{R}}_{\mathit{S}\mathit{t}\mathit{a}\mathit{r}}=8{\mathit{ }\mathit{x}\mathit{ }\mathit{R}}_{\mathit{S}\mathit{u}\mathit{n}}=5.57\mathit{ }\mathit{x}\mathit{ }{10}^9\mathit{ }\mathit{m}).$ Then, the jet of plasma particles are accelerated protons and the dominant plasma ions, they escaped from an external surface of this massive star with a speed $( \mathit{v}=\sqrt{{\displaystyle \frac{2\mathit{G}\mathit{ }\mathit{M}\mathbf{ }}{\mathit{r}}}}=977\mathit{ }{\displaystyle \frac{\mathit{K}\mathit{m}}{\mathit{s}}}\mathit{ }).$ Only particles with a speed over (977 km/s) have the capacity to escape from this massive star without additional magnetic or radiative driving. The jets below ($\mathit{v}=977\mathit{ }\mathit{k}\mathit{m}/\mathit{s}$) are partially bound, and jets above this speed freely escaping plasma. Blue star, with a mass 20 times the mass of the Sun, requires a speed of 1000 kilometers per second and a temperature of 40 million kelvins to launch freely escaping lightning-driven plasma jets. The jet speed of proton particles are ($\mathit{v}=100\mathit{ }\mathit{k}\mathit{m}/\mathit{s}$), the mass of a proton particle is $(\mathit{ }{\mathit{m}=\mathit{m}}_{\mathit{P}}={1.6726\mathit{ }\mathit{x}\mathit{ }10}^{-27}$ kg), then the temperature of a hydrogen atom or proton particles on the surface of a massive star is very low, it is approximately $\left(\mathit{ }\mathit{T}={\displaystyle \frac{\mathit{m}{\mathit{v}}^2}{3{\mathit{K}}_{\mathit{B}}}}={4\mathit{ }\mathit{x}\mathit{ }10}^5\mathit{ }\mathit{K}\mathit{ }\right).$ The jet speed of proton particles are ($\mathit{v}=1000\mathit{ }\mathit{k}\mathit{m}/\mathit{s}$), the mass of a proton particle is $(\mathit{ }{\mathit{m}=\mathit{m}}_{\mathit{P}}={1.6726\mathit{ }\mathit{x}\mathit{ }10}^{-27}$ kg), then the temperature of proton particles on the surface of a blue star is very high, it is approximately $\left(\mathit{ }\mathit{T}={\displaystyle \frac{\mathit{m}{\mathit{v}}^2}{3{\mathit{K}}_{\mathit{B}}}}={4\mathit{ }\mathit{x}\mathit{ }10}^7\mathit{ }\mathit{K}\mathit{ }\right).$ The jet speeds of ions and proton particles are lower than 977 kilometers per second as compared to escape velocity of the energetic particles may reach thousands of kilometers per second.

Figure 1. Plasma Jet speed compared to escape velocity.

Figure 1. Plasma Jet speed compared to escape velocity.

The proton particles and ions have variable speeds vs Temperature since the powerful plasma jets and stellar lightnings (lightning trees) are coming out rapidly from internal surface of a blue star. The speed and temperature of proton particles had been increased rapidly, typically over escape velocity point. The lightning tree has maximum speed and temperature. The stars have huge mass, gravity, and intense electromagnetic field. The particles with lower speed and minimum temperature may fall back on the surface of star as a plasma rain. The plasma jets and stellar lightnings with higher speed, energy, momentum, and maximum temperature may escape into space as coronal mass ejection and cosmic radiations. The plasma jets and lightning trees are main source of heating solar corona and stellar corona. The coronal heating dynamic is activated by powerful plasma jets and lightning trees. The plasma jets and stellar lightnings together may participate to enhance turbulences in the surface of stars to accelerate particles and ions, and release high-energy photon particles as gamma rays and x-rays. Nuclear fusion takes place in the heart of stars, where light atoms fused to produce heavy atomic nuclei, and releasing energetic particles, neutrino, and electromagnetic radiations. Photon particles required longer time to come out from depth of stars. The maximum number of photon particles trapped in the core of stars is involved to enhance their central temperature, interior pressure, and accelerate nuclear fusion processes. Nuclear fusion is a main source of the plasma jets, lightning trees, stellar lightnings, photon particles, neutrino particles, coronal mass ejection, cosmic radiations, and coronal heating dynamics.

Figure 2. Plasma jets’ speed vs particle temperature.

Figure 2. Plasma jets’ speed vs particle temperature.

3. Nuclear Fusion Effects

The surface temperature and luminosity of stars have increased violently while they are burning the maximum amount of nuclear fuel. The nuclear fusion of stars acts directly on the formation and evolution of stars. It is involved in the formation of plasma jets, stellar lightnings, stellar flares, coronal mass ejection and increases the surface temperature, luminosity, and coronal heating dynamics. The visible Universe has enough mass, gravity, hydrogen fuel, stellar nebulae, and powerful black holes to form low-mass and high-mass stars. We are living in a professional and intelligent Universe. The Universe can create, develop and destroy itself. The visible Universe was a superparticle that could be created from compressed amount of matter inside a huge parent black hole and escaped into infinite space outside an event horizon of a supergiant black hole, evaporating to form all galaxies, celestial objects, and stars. The Universe is intelligent enough to make stars, chemical elements, and the human body from star dust. We are survived superparticle, star dust and grandchildren of stellar remnants due to supernova events. Low-mass stars, like the Sun, can fuse hydrogen nuclei to form helium. In the heart of massive stars, heavy chemical elements had been formed due to nucleosynthesis. The central temperature and central pressure of stars are increased violently through the burning of multiple types of chemical elements, typically in massive stars. About half of the total mass of a star is compressed into its core, combined, and squeezed into the 4 percent of the total radius of the star to fuse light and Heavy chemical elements [15]. The star is a big ball of the hottest gas and plasma; as a result, most of the mass of the star falls into the center to ignite nuclear fusion. The central radius of a star is only four percent of its total radius, and half of its mass is condensed into such a small fusion ball to protect stars from explosion, and store the maximum amount of thermal energy, and increase both the stellar central temperature and central pressure.

Problem 2: The main-sequence star or the high-mass star with a mass of 20 times the mass of a Sun $(3.98 x {10}^{31} kg)$, and its radius 8 times the radius of a Sun $( 5.57 x {10}^9 m)$, the central mass of a star is half of its total mass, and the central radius of a star is four percent of its total radius to fuse heavy elements, calculate its central radius, central mass, maximum central pressure, central temperature, minimum distance between two atomic nuclei to fuse, speed of hydrogen or proton particles, DE Broglie wavelength of hydrogen particles and its frequency ?.

Solution:

${\mathit{R}}_{\mathit{c}}=0.04\mathit{ }\mathit{x}\mathit{ }\mathit{R}=0.04\mathit{ }\mathit{x}\mathit{ }{5.57\mathit{ }\mathit{x}\mathit{ }10}^9\mathit{ }\mathit{m}=222800000\mathit{ }\mathit{m}\mathit{ }.$${\mathit{R}}_{\mathit{c}}=222800\mathit{ }\mathit{k}\mathit{m}.$

where ${\mathit{R}}_{\mathit{c}}$ is the central radius of a fusion ball, it is about four percentage of the total radius of a star, where half percent of the total mass of a massive star is compressed there to fuse heavy chemical elements under huge central pressure and temperature:

${\mathit{M}}_{\mathit{c}}=0.5\mathit{ }\mathit{x}\mathit{ }\mathit{M}=0.5\mathit{ }\mathit{x}\mathit{ }3.98 x {10}^{31} kg=1.99 x {10}^{31} kg\mathit{ }.$where ${\mathit{M}}_{\mathit{c}}$ is the Central mass of a massive star, it is about half of the total mass of star, because the star is a huge gaseous sphere of condensed plasma. To initiate a nuclear fusion continuously in the core of star, the star should compress most of its mass at its heart. The nuclear fusion reaction is required massive stars, huge central pressure and maximum central temperature.

${\mathit{\rho }}_{\mathit{c}}={\displaystyle \frac{{\mathit{M}}_{\mathit{c}}\mathbf{ }}{\frac{4}{3}\mathit{ }\mathit{\pi }\mathit{ }{{\mathit{R}}_{\mathit{c}}}^3}}={\displaystyle \frac{{\mathit{M}}_{\mathit{c}}\mathbf{ }}{\frac{4}{3}\mathit{ }\mathit{\pi }\mathit{ }{\left(0.04\mathit{ }\mathit{x}\mathit{ }\mathit{R}\right)}^3}}={\displaystyle \frac{1.99 x {10}^{31} kg}{\frac{4}{3}\mathit{ }\mathit{\pi }\mathit{ }{\left(222800000\mathit{ }\mathit{m}\right)}^3}}$  .

${\mathit{\rho }}_{\mathit{c}}=429773\mathit{ }{\displaystyle \frac{\mathit{k}\mathit{g}}{\mathit{ }{\mathit{m}}^3}}$  .

where ${\mathit{\rho }}_{\mathit{c}}$ is the central density of a nuclear fusion ball.

${\mathit{P}}_{\mathit{c}}=\left(\mathit{ }{\displaystyle \frac{3\mathit{G}\mathit{ }{{\mathit{M}}_{\mathit{c}}}^2}{8\mathit{\pi }\mathit{ }{{\mathit{R}}_{\mathit{c}}}^4}}\mathit{ }\right)=\left(\mathit{ }{\displaystyle \frac{3\mathit{ }\mathit{x}\mathit{ }6.674\mathit{ }\mathit{x}\mathit{ }{10}^{-11}\mathrm{ }\frac{{N. m}^2}{{kg}^2}\mathit{ }\mathit{x}{\mathit{ }\left(1.99\mathit{ }\mathit{x}\mathit{ }{10}^{31}\mathit{ }\mathit{k}\mathit{g}\right)}^2}{8\mathit{\pi }\mathit{ }(222800000\mathit{ }\mathit{m}{)}^4}}\mathit{ }\right).$${\mathit{P}}_{\mathit{c}}=1.28\mathit{ }\mathit{x}\mathit{ }{10}^{18}{\displaystyle \frac{\mathit{N}}{{\mathit{m}}^2}}\mathit{ }.$

The maximum central pressure ${\mathit{P}}_{\mathit{c}}$ is determined by using an equation of hydrostatic equilibrium in the core of a blue star.

${\mathbf{T}}_{\mathbf{c}}={\displaystyle \frac{\mathit{ }\mathit{\mu }\mathit{ }{\mathit{m}}_{\mathit{H}}{\mathbf{ }\mathbf{P}}_{\mathbf{c}}}{{{\mathit{\rho }}_{\mathit{c}}\mathit{K}}_{\mathit{B}}}}={\displaystyle \frac{\mathit{ }\left(0.61\mathit{ }\mathit{x}\mathit{ }1.674\mathit{ }\mathit{x}\mathit{ }{10}^{-27}\mathit{ }\mathit{k}\mathit{g}\mathit{ }\mathit{x}\mathit{ }1.28\mathit{ }\mathit{x}\mathit{ }{10}^{18}\mathit{ }\mathit{P}\mathit{a}\right)}{\left(\mathit{ }429773\mathit{ }\frac{\mathit{k}\mathit{g}}{\mathit{ }{\mathit{m}}^3}\mathit{ }\mathit{x}\mathit{ }1.38\mathit{ }\mathit{x}\mathit{ }{10}^{-23}\mathit{ }\frac{\mathit{J}}{\mathit{K}}\mathit{ }\right)\mathit{ }}}$${\mathbf{T}}_{\mathbf{c}}=220382469\mathit{ }\mathit{K}.$The mean molecular weight for the ionised gas the hydrogen and helium atoms $( \mathit{\mu }=0.61)$. Where ${\mathbf{T}}_{\mathbf{c}}$ is the maximum central temperature of proton particles and ions in the core of massive Stars by using an equation of ideal gas in hydrostatic balance of the main sequence stars.

${\mathit{r}}_{\mathit{m}\mathit{i}\mathit{n}}$

=${\mathit{K}}_{\mathit{C}}{\displaystyle \frac{{{\mathit{Z}}_1{\mathit{ }\mathit{Z}}_{2\mathit{ }}\mathit{e}}^2}{\mathbf{ }\frac{3}{2}\mathit{ }{\mathit{K}}_{\mathit{B}}{\mathbf{T}}_{\mathbf{c}}}}=\left(\mathit{ }8.99\mathit{ }\mathit{x}\mathit{ }{10}^9\mathit{ }{\displaystyle \frac{{\mathit{N}.\mathit{m}}^2}{{\mathit{c}}^2}}\mathit{ }\mathit{x}\mathit{ }{\displaystyle \frac{{1\mathit{ }\mathit{x}\mathit{ }1\mathit{ }\mathit{x}\mathit{ }\left(1.6\mathit{ }\mathit{x}{10}^{-19}\mathit{ }\mathit{C}\mathit{ }\right)}^2}{\mathbf{ }\frac{3}{2}\mathit{ }\mathit{x}\mathit{ }1.38\mathit{ }\mathit{x}\mathit{ }{10}^{-23}\mathit{ }\frac{\mathit{J}}{\mathit{K}}\mathit{ }\mathit{x}\mathit{ }220382469\mathit{ }\mathit{K}\mathit{ }}}\right).$${\mathit{r}}_{\mathit{m}\mathit{i}\mathit{n}}={5.044\mathit{ }\mathit{x}\mathit{ }10}^{-14}\mathit{ }\mathit{m}.$Assuming in proton—proton fusion processes ${(\mathit{Z}}_1{=\mathit{ }\mathit{Z}}_{2\mathit{ }}=1)$, and $(\mathit{e}=1.6\mathit{ }\mathit{x}{10}^{-19}\mathit{ }\mathit{C}\mathit{ })$ is an electric charge. Then, ${\mathit{r}}_{\mathit{m}\mathit{i}\mathit{n}}$ is the minimum distance between any atomic nuclei to fuse, and it is crucial to produce heavy elements, and release huge amount of nuclear energy, photon particles, neutrino particles, and energetic particles. Most photon particles are trapped in the heart of a star to increase its internal temperature. The neutrino particles may come out from the heart of stars for few seconds due to their huge power, maximum energy, easy penetration, high frequency, minimum radius, narrow wavelength, less interaction with matter, and rapid penetration through celestial objects except the heart of black holes. At every second, millions of tons of stellar fuel fuse into new heavy matter, and releasing huge amounts of energy into space. About 97% of the gravitational energy of the star is converted into neutrino particles in the supernovae event, and 3% may convert into photons, atoms, subatomic particles, and fabriton particles in the structure of an exploded stars. Most of the gravitational energy of the stars is converted into neutrino particles in the collapsed stars, and a small portion of their energy is dissipated as vibrational energy for exploded stars. In main sequence stars, only few ratio of the gravitational energy is converted into neutrino particles during nuclear fusion processes.

At the beginning, I have to calculate the speed, wavelength, and frequency of protons or hydrogen ions, and much necessary to find the distance between two protons or two hydrogen ions in the core of massive stars:

$\mathit{v}=\sqrt{{\displaystyle \frac{3{\mathit{K}}_{\mathit{B}}\mathit{ }{\mathbf{T}}_{\mathbf{c}}}{{\mathit{m}}_{\mathit{H}}}}}=\sqrt{{\displaystyle \frac{\left(\mathit{ }3\mathit{ }\mathit{x}\mathit{ }1.38\mathit{ }\mathit{x}\mathit{ }{10}^{-23}\mathit{ }\frac{\mathit{J}}{\mathit{K}}\mathit{ }\mathit{x}\mathit{ }220382469\mathit{ }\mathit{K}\right)\mathit{ }}{\left(1.674\mathit{ }\mathit{x}\mathit{ }{10}^{-27}\mathit{k}\mathit{g}\right)}}}$ .

Then, the speed of hydrogen ion or proton particle has been determined is $({\mathit{v}}_{\mathit{H}}=2334591.85\mathrm{ }{\displaystyle \frac{\mathrm{m}}{\mathrm{s}}}).$ Then, $\mathit{v}$ is the thermal velocity of particles, specifically hydrogen ions, $({\mathit{m}}_{\mathit{H}}={\mathit{m}}_{\mathit{P}\mathit{r}\mathit{o}\mathit{t}\mathit{o}\mathit{n}})$ is the mass of hydrogen ion equals to the mass of proton particle.

$\mathit{\lambda }$ =${\displaystyle \frac{\mathit{h}}{\mathbf{ }\mathbf{m}\mathit{v}}}={\displaystyle \frac{\mathit{h}}{\mathit{ }{\mathit{m}}_{\mathit{H}}.{\mathit{v}}_{\mathit{H}}}}={\displaystyle \frac{(\mathit{ }6.63\mathit{ }\mathit{x}{10}^{-34}\mathit{ }\mathit{k}\mathit{g}.\frac{{\mathit{m}}^2}{\mathit{s}})}{(1.674\mathit{ }\mathit{x}\mathit{ }{10}^{-27}\mathit{k}\mathit{g}\mathit{ }\mathit{x}\mathit{ }2334591.85\mathrm{ }\frac{\mathrm{m}}{\mathrm{s}} )}}={1.696\mathit{ }\mathit{x}\mathit{ }10}^{-13}\mathit{ }\mathit{m}$.

${\mathit{\lambda }}_H={1.696\mathit{ }\mathit{x}\mathit{ }10}^{-13}\mathit{ }\mathit{m}.$ The de Broglie wavelength of the proton particle in the core of a massive star has been determined as $({\mathit{\lambda }}_H=0.1696 \mathrm{p}\mathrm{m})$.

By using a speed equation of particles and an electromagnetic waves to calculate the frequency of the vibrated hydrogen ions and protons particles: $\mathit{v}=\mathit{f}\mathit{ }\mathit{\lambda }$.

Rearrange above equation to determine the frequency of oscillating hydrogen ions or proton particles:$\mathit{ }\mathit{f}={\displaystyle \frac{\mathit{v}}{\mathit{\lambda }}}={\displaystyle \frac{( 2334591.85\mathrm{ }\frac{\mathrm{m}}{\mathrm{s}})}{{(1.696\mathit{ }\mathit{x}\mathit{ }10}^{-13}\mathit{ }\mathit{m})}}={1.376\mathit{ }\mathit{x}\mathit{ }10}^{19}\mathit{ }\mathit{H}\mathit{z}.$At first step, I need to find number density of hydrogen or proton Particles$\mathit{ }\mathit{n}$ to calculate the minimum distance between two hydrogens or two proton particles:

$\mathit{n}={\displaystyle \frac{{\mathit{\rho }}_{\mathit{c}}}{{\mathit{m}}_{\mathit{H}}}}={\displaystyle \frac{\left(429773\mathit{ }\frac{\mathit{k}\mathit{g}}{\mathit{ }{\mathit{m}}^3}\right)}{1.674\mathit{ }\mathit{x}\mathit{ }{10}^{-27}\mathit{k}\mathit{g}}}=2.567\mathit{ }\mathit{x}\mathit{ }{10}^{32}\mathit{ }{\displaystyle \frac{\mathit{P}\mathit{a}\mathit{r}\mathit{t}\mathit{i}\mathit{c}\mathit{l}\mathit{e}\mathit{s}}{{\mathit{m}}^3}}$  .

$\mathit{n}=2.567\mathit{ }\mathit{x}\mathit{ }{10}^{32}\mathit{ }{\displaystyle \frac{\mathit{P}\mathit{a}\mathit{r}\mathit{t}\mathit{i}\mathit{c}\mathit{l}\mathit{e}\mathit{s}}{{\mathit{m}}^3}}.$Then, substitute the value of $\mathit{n}$ into this equation $(\mathit{n}={\displaystyle \frac{\mathit{N}}{\mathit{V}}}={\displaystyle \frac{1}{\mathit{V}={\mathit{d}}^3}}\mathit{ })$ to calculate (d) the distance between two hydrogen ions or two protons: $\left(2.567\mathit{ }\mathit{x}\mathit{ }{10}^{32}{\displaystyle \frac{\mathit{P}\mathit{a}\mathit{r}\mathit{t}\mathit{i}\mathit{c}\mathit{l}\mathit{e}\mathit{s}}{{\mathit{m}}^3}}={\displaystyle \frac{1}{{\mathit{d}}^3}}\mathit{ }\right).$$\mathit{d}={\displaystyle \frac{1}{\sqrt[3]{\mathit{n}}}}={\displaystyle \frac{1}{\sqrt[3]{2.567\mathit{ }\mathit{x}\mathit{ }{10}^{32}\mathit{ }\frac{\mathit{P}\mathit{a}\mathit{r}\mathit{t}\mathit{i}\mathit{c}\mathit{l}\mathit{e}\mathit{s}}{{\mathit{m}}^3}}}}$ . $\mathit{d}=1.573\mathit{ }\mathit{x}\mathit{ }{10}^{-11}\mathit{ }\mathit{m}.$

4. Plasma Jets and Stellar Lightnings

Massive stars drive some of the most violent phenomena in the universe, characterized by high-velocity plasma ejections and extreme electrical activity, or stellar lightnings during formation, fusion, evolution, and ultimate death of stars. Plasma jets are well-documented phenomena that emanate from massive stars, particularly during their birth and, in some cases, throughout their evolution, and death as supernovae, when they form stellar-mass black holes. These jets and lightning trees are highly collimated streams of ionized matter (plasma) that shoot out from the poles of massive stars or their remnants at near-light speeds. Powerful Magnetic fields of the stars, typically massive stars, may ionize gas into twin plasma jets that can stretch out for thousands kilometers, scaling in size with the star’s mass. Twin plasma jets are two massive plasma jets of highly energetic particles that come out from an external surface of a massive star at the same time. Massive, and hot young stars (some over seven times the mass of our Sun) emit intense stellar lightnings and radiation that make the surrounding gas glow. High-energy X-rays from newborn stars and massive stars appear as sparkling. Massive stars, such as those with masses above 8 to 200 solar masses, produce powerful line-driven, hottest winds during their evolution, losing mass at high rates. Plasma jets, spicules, and radiation, flares, and energetic particles come out from low-mass stars and massive stars [16–22]. Nuclear fusion in massive stars with a mass of 20 times the solar mass powers their evolution by converting lighter elements into heavier ones. The stellar lightnings are a rapid explosion of highly energetic particles and ions that are displayed as huge trees on the surface of stars, and enhance the turbulence in the photosphere and chromosphere zones to heat them and release coronal loops, spicules, and coronal mass ejections.

In the deep core of massive stars, an onion-like structure of burning shells is created. The nuclear fusion process starts with hydrogen fusing to form helium. Nuclear fusion can fuse light atoms to create heavy elements, releasing energy and energetic particles. About 700 million tons of hydrogen atoms are fused at any second in the heart of solar mass stars to form 695 million tons of helium, releasing 5 million tons as energy. The mass of fused elements could be higher than the mass of the produced elements in the nuclear fusion processes due to the dissipation fraction of element mass into energy in the nuclear fusion processes. Nuclear fusion processes in the heart of massive stars are violent and powerful compared to those of low-mass stars. In fact, massive stars can fuse the maximum amount of mass and multiple elements at once to produce additional energy and form new heavy elements. Chemical elements are formed abundantly in the heart of massive stars, and powerful plasma jets are displayed on their surface. An additional mass of star, maximum gravitational energy of the star, continuous nuclear fusion processes in the core of a star, and an accumulation of photon particles there in the lower shells of stars may increase the central temperature, thermal energy, and central pressure of the star to start nuclear fusion processes widely and produce new heavy elements, typically in the core of a massive star. In the heart of massive stars, helium, carbon, neon, oxygen, Germanium, magnesium, and silicon are fused under violent conditions, and even thermal energy and pressure culminate in the hottest iron core, which leads to the expansion of an outer shell of the massive stars or lead to a supernova event. Most chemical elements in the periodic table, and those in the structure of living creatures or celestial objects, could be formed in the heart of massive stars, and those elements are distributed into space after a supernova event. A supernova event is the violent death and explosion of a massive star. It occurs after the star’s nuclear fuel is exhausted and its hydrostatic balance has vanished entirely. About half of the mass of a massive star is concentrated in a nuclear fusion ball or around it to trap as many photon particles as possible, enhance thermal energy, and increase the star’s central pressure and temperature, which leads to enhanced nuclear fusion processes and produces new heavy elements. The star is a big ball of the hottest plasma, and its volume changes due to nuclear fusion processes and the release of energy. The galaxy contains millions to billions of main-sequence stars that are burning their nuclear fuel to feed the universe with additional energy and chemical elements. The thermal energy, central pressure, central temperature, external surface temperature, luminosity, and total volume of the stars are dynamically changeable, providing enough opportunity for stars to release photons, energy, stellar lightnings, and plasma jets into space. The stars can expand out or shrink inward since their internal energy or pressure increases or decreases due to the continuous nuclear fusion process in the heart of stars.

$\mathit{V}=\mathit{A}\mathit{d}={\displaystyle \frac{4}{3}}\mathit{\pi }{\mathit{r}}^3$ (7)

$\mathbf{W}=\mathit{F}\mathit{d}={\int }_{{\mathit{V}}_{\mathit{i}}}^{{\mathit{V}}_{\mathit{f}}}\mathit{P}\mathit{d}\mathit{V}=\mathit{P}({\mathit{V}}_{\mathit{f}}-{\mathit{V}}_{\mathit{i}})$

 (8)

where W is the work done by a star, F is a force, d is a displacement, P is a pressure, A is the surface area of the star, V is the volume of the star, and r is the radius of spherical stars. The radius r and volume of star dV are changeable due to an increment and decrement in the temperature, pressure, and size of star. Indeed, any change in the size of a star, dV, multiplied by a pressure P that is equalized to the work done by a star, which is released as stellar luminosity, plasma jets, stellar lightnings, and coronal mass ejections, or converted into energy that is lost every second as luminosity and brightness. The internal energy and thermal energy in the heart of a star have increased due to the sustainable nuclear fusion processes. The size of a star is changeable due to an increase and decrement of the central pressure and central temperature of the star. An external shell of a star is expanding due to an increase in the central pressure and temperature of the star. Powerful plasma jets and lightning trees may come out rapidly from the deepest point of a massive star due to the expansion and weakening of the external shell of a star. The lightning tree of ions and energetic particles may come out from the inner shells of a massive star. The stellar lightnings may come out as trees from the lower shells of a star. The stellar lightnings are several ions and energetic particles that may come out from massive stars and explode as a huge tree on the surface of stars. The external surface of stars may rise as a hill and blow up to emanate a stellar lightning as a huge tree of the hottest and energetic plasma particles. The lifetime of a stellar lightning is a few seconds. The lightning tree may come out from the inner shell of stars to expand vertically and annihilate on its surface at the shortest time as compared to the lifetime of a powerful plasma jets. The plasma jets contained a large number of energetic particles that may come from the inner shells of stars and explode as hydrogen bombs. The plasma jets have a longer lifetime and contained huge amount of accelerated particles. They can persist on the surface of a massive star for hours, days, or weeks. The life time of stellar lightning is few seconds since it is coming out urgently from deep point of the star and annihilated on its surface after evaporation. The stellar lightning may come out vertically as a thinner, denser and powerful string of plasma jet from lower shell of a massive star, and annihilated immediately on its external surface as branches of huge tree due to powerful gravitational field and electromagnetic field of star. The stellar plasma jet may come out as a huge explosion of the volcanic or hydrogen bomb plasma that stretched out thousands kilometers on the surface of star and curved on its surface again due to gravity and electromagnetic field of star. The plasma jets, solar flare, spicules, coronal mass ejects and stellar lightnings are hottest photon and energetic plasma particles may display on the surface of stars, and they could be curved, and dropped back on the surface of stars as plasma rain are fall down to heat up an external surface of stars. The plasma jet may come out from the depth of a massive star and explode as a huge volcanic with a power of millions of hydrogen bombs when explode at same moment at same position. Both of plasma jets and lightning trees participated in the coronal heating dynamic. The lower branch of the plasma jet, with a blue colour, is much hotter than the top portion of the jet, which is evaporated and scattered in yellow and red colours. The lower portion of the lightning tree or plasma jet contains condensed, energetic, and hottest plasma particles. The upper portion of the lightning tree or plasma jet could be stretched out, evaporated, and lose most of its thermal energy and its kinetic energy after coming away from the surface of the stars. The stars are surrounded by a powerful electromagnetic field, and a huge gravitational field of the very distorted and condensed fabriton particles. The photon particles and plasma particles are struggling to leave the external surface of a star due to the powerful gravity and the electromagnetic field of the star. Dark fabric matter and energy of the fabriton particles could be distorted and compressed around massive stars, as a result these stars have a powerful gravitational field and are induced to burn their fuel quickly as compared to low-mass stars. Indeed, the central pressure and temperature of the massive stars are increased due to nuclear fusion processes in the heart of massive stars that lead to expand outer shells of the stars and release powerful plasma jets and stellar lightnings into space over the surface of the stars.

Figure 3. Powerful Plasma Jets and Stellar Lightnings Come Out from a Massive Star.

Figure 3. Powerful Plasma Jets and Stellar Lightnings Come Out from a Massive Star.

5. Coronal Heating Dynamics

The solar corona extends above the photosphere and chromosphere, reaching to the edge of the solar atmosphere, where it merges with the solar wind. It is the hottest zone of the solar atmosphere and contains mostly accelerated particles that are moving with extreme speed and that radiate photons with higher energy. The solar corona is the hottest, tenuous, outermost layer of the Sun’s atmosphere, extending from the chromosphere’s top (above the photosphere) out to 5 million kilometers above the photosphere. It is characterized by mostly ionized and energetic particles with a million-degree plasma, which merges with the solar wind at the irregular Alfvén surface. It is defined as a thin transition region that separates it from the cooler chromosphere, and it is a most visible during total solar eclipses or with solar equipment that tested at this time. The solar corona is formed by strange structures such as prominences, coronal loops, and helmet streamers. The solar corona is shaped by a complex interplay of magnetic field gradients, temperature variations, gravitational field effects of the stars, and variable plasma density from the interior structure of the stars and their surface. Scientists proposed that these structures are all fundamentally sculpted by the powerful Sun’s magnetic fields, which act as “highways” for charged particles and energy. Coronal Loops are ropey, curving strands of plasma that follow closed magnetic field lines connecting different magnetic regions on the solar surface. They are displayed abundantly during solar maximum and are often found near sunspots and active regions of the Sun. While they glow for weeks, some change rapidly and are associated with the rapid display of solar flares. Solar Prominences (and Filaments) are enormous, cool, and dense plasma structures anchored in the photosphere and can persist for months, sometimes erupting and escaping into space as Coronal Mass Ejections (CMEs) and entering Earth’s magnetic field which glows as nice colours of polar aurora. Helmet Streamers or coronal streamers, are large, cap-like structures with long, pointed peaks of the plasma and accelerated particles that are usually available in sunspots and active regions [23–26]. They are formed by closed intense magnetic loops that are trapping dense coronal gases. The concept of coronal heating dynamics has been expanded recently in the field of solar physics by using astronomical equipment.

5.1. Primary Heating Mechanisms

Scientific consensus and numerical models indicate that the energy required to heat the solar corona (up to millions of degrees) is indeed provided by the intense solar magnetic field, which is constantly twisted, funnelled, warped, and violently turbulent, and is entangled with turbulent convective motions in the photosphere. Magnetic reconnection, often referred to as DC heating (direct current heating) in the context of coronal heating, is a fundamental plasma process where magnetic field lines break and reconnect, releasing stored magnetic energy as heat and kinetic energy [27–30]. The nanoflares are impulsive energy releases that, individually, are tiny but collectively heat the entire corona. This process is widely considered a leading candidate for heating the solar corona to its multimillion-degree temperatures. MHD Wave Heating (AC Heating): Magnetohydrodynamic (MHD) waves, such as Alfvén waves, carry thermal energy as the hottest and most energetic particles from the solar interior and release it into the corona. This energy is dissipated into heat through processes like resonant absorption (waves transferring energy to specific plasma layers) and phase mixing (waves on adjacent field lines becoming out of sync and creating friction). Recent research in solar physics suggests that magnetohydrodynamic (MHD) waves and magnetic reconnection are not isolated events, but rather intricately linked through a “Symbiosis of Waves and Reconnection (SWAR)”. Symbiosis (SWAR): Modern research suggests these mechanisms are linked; waves can trigger reconnection by collapsing magnetic null points, while reconnection events can launch new waves. This interconnected framework indicates that waves can trigger reconnection, while reconnection events simultaneously launch new waves and turbulences.

5.2. Observed Dynamics and Structures

The coronal heating problem is still a mystery of why the Sun’s outer atmosphere (the solar corona) is millions of degrees Celsius hot, as compared to the lower surface temperature of the solar surface in the photosphere and chromosphere are 5000 or 10000 degrees Celsius. Observed dynamics and structures indicate that the corona is not heated uniformly but through highly dynamical processes, impulsive phenomena, and spatially resolved, that is, observed and multi-scale magnetic processes. Coronal Loops: These are closed magnetic structures that trap hot plasma and solar energetic particles [31,32]. They often show impulsive heating, where they brighten in gamma rays and X-rays at first moment and then cool down rapidly through various Extreme Ultraviolet (EUV) wavelengths or visible light. Coronal Rain: the Coronal rain is a phenomenon in the solar corona where cool, dense plasma condensations form and fall along magnetic field lines, typically in active region coronal loops [33]. When heating is concentrated at the loop foot points, it can cause thermal instability, leading to the catastrophic cooling of plasma that falls back directly toward the surface as “plasma rain”. Solar Wind: The solar wind consists of ionized plasma particles such as electrons, protons, and helium ions [34–40]. It is open magnetic field lines in coronal holes allow the hottest plasma state and energetic particles to escape, forming the fast solar wind. Coronal holes are low-density, cool, and dark regions on the Sun where open, unipolar magnetic field lines allow plasma to escape freely into space, forming the high-speed solar wind, which reaches speeds of 700 km/s to 850 km/s. The Coronal holes act as funnels for mainly the solar wind, and abundant during solar minimum. Accelerated by wave-driven heating, this fast solar wind originates from the Sun’s polar regions or low-latitude coronal holes.

6. Stars Change Size

Stars change size throughout their central pressure, temperature change, and lifecycles, typically expanding into red giants as they exhaust hydrogen fuel, increasing their radius by hundreds of times. This expansion occurs as the core collapses and the outer shell burns hotter, causing the star’s outer layers to swell and cool [41–45]. Ultimately, intermediate stars become white dwarfs, while massive stars end as neutron stars or supernovae to give birth for a newborn stellar mass black hole. The volume of a star changeable dynamically due to work done by central pressure of stars. The star’s volume must change to supply the energy it radiates, assuming that energy comes from work done by pressure.

Problem 3: The massive star with a mass of 20 times the mass of a Sun $(3.98 x {10}^{31} kg)$, and its radius 8 times the radius of a Sun $( 5.57 x {10}^9 m)$, the central pressure in the heart of this massive star is $({\mathit{P}}_{\mathit{c}}=1.28\mathit{ }\mathit{x}\mathit{ }{10}^{18}{\displaystyle \frac{\mathit{N}}{{\mathit{m}}^2}}\mathit{ })$, and thermal energy that lost per a second as a luminosity is $\left(\mathit{W}=\mathit{E}=1.38\mathit{ }\mathit{x}\mathit{ }{10}^{31}\mathit{ }\mathit{J}\mathit{ }\right),$ calculate the volume that change in the volume of star?.

Solution: by using equation (8) to calculate any change in the size of a star due to work done by central pressure or thermal energy of the massive Star.

$\mathrm{W}=PdV, \mathbf{d}\mathbf{V}={\displaystyle \frac{E}{P}}={\displaystyle \frac{\left(1.38\mathit{ }\mathit{x}\mathit{ }{10}^{31}\mathit{ }\mathit{J}\mathit{ }\right)}{\left(1.28\mathit{ }\mathit{x}\mathit{ }{10}^{18}\frac{\mathit{N}}{{\mathit{m}}^2}\right)\mathit{ }}}=1.08\mathit{ }\mathit{x}\mathit{ }{10}^{13}\mathit{ }{\mathit{m}}^3\mathit{ }.$ Where dV is the tiny change in the volume of a star due to an energy radiated per 1 second. Over millions of years, this would accumulate into a large contraction unless nuclear energy dominates. Take a massive star with a radius 8 times the Sun and its total volume is incredible high about$\mathit{V}={\displaystyle \frac{4}{3}}\mathit{\pi }{\mathit{r}}^3=7.23\mathit{ }\mathit{x}\mathit{ }{10}^{29}\mathit{ }{\mathit{m}}^3.$ This fractional change in the volume of a massive star per second has correct physical interpretation. $( {\displaystyle \frac{dV}{V}}={\displaystyle \frac{\left(1.08\mathit{ }\mathit{x}\mathit{ }{10}^{13}\mathit{ }{\mathit{m}}^3\mathit{ }\right)}{\left(7.23\mathit{ }\mathit{x}\mathit{ }{10}^{29}\mathit{ }{\mathit{m}}^3\right)\mathit{ }}}=1.49 x {10}^{-17}).$ This result nicely illustrates a classic stellar-physics insight: Because core pressures are enormous, even tiny contractions could in principle, power huge luminosities. But stars avoid catastrophic contraction because fusion replaces gravitational energy loss. Here is the requested information regarding stellar structure and energy generation. Stellar Radiation Sources: Fusion vs. ($W=PdV$) Work, Stars do not radiate mainly via work (bulk mechanical contraction/expansion). Main-Sequence Phase: Stars shine primarily by producing energy through nuclear fusion in their cores. This fusion creates the outward pressure necessary to counteract gravity, allowing the star to exist in a stable hydrostatic equilibrium, rather than relying on contraction. When $W=PdV$ is Relevant: Mechanical work (gravitational contraction/expansion) is the primary energy source during Pre-main-sequence contraction while the protostar is gathering mass and heating up before hydrogen fusion ignites. Late evolutionary phases: when a star exhausts its core fuel and collapses, leading to shell burning. Kelvin–Helmholtz Timescale: This timescale represents the time a star can remain luminous based solely on gravitational contraction ($PdV$= work) rather than nuclear fusion. Pressure Structure: Not Uniform Pressure within a star is not uniform; it is a function of radius, P(r). Radial Dependence: Pressure, density, and temperature vary by many orders of magnitude from the center of the star to its surface. Order of Magnitude Estimates: Using a single pressure value (such as a mean pressure) for a star is only an order of magnitude estimate (e.g., using the Virial Theorem to find mean pressure). Detailed stellar models require solving equations of stellar structure, which include the pressure gradient, ${\displaystyle \frac{dp}{dr}}$ to account for the actual change in pressure with radius.

7. Surface Temperature and Luminosity of a Main Sequence Star

The surface temperature of a Sun or a massive star could be increased rapidly due to an increment in the nuclear fusion processes, powerful plasma jets, stellar lightnings, sunspots, and solar flare display on the surface of most stars. About half of the mass of a main-sequence star is contracted in its core to accelerate the nuclear fusion processes and produce multiple chemical elements. The nuclear fusion process in the heart of the Sun may fuse hydrogen isotopes to form a helium atom, releasing enough energy into space. The photon particles require thousands of years to come out from the deepest point in the heart of a Sun or stars after many collisions, absorptions, reflections, scattering, and transitions with matter and subatomic particles in the structure of a Sun or Stars. Most photon particles are trapped and opaque in the inner shells of stars and are involved directly in the burning of heavy elements since central temperature of the stars has been increased widely. The tussle between thermal pressure and gravity enhanced the stability of stars, typically main-sequence stars. The main-sequence stars play a crucial role in the formation and evolution of stars, and most of the chemical elements are formed in this stage of a stellar lifetime. The volume or the radius of a fusion ball in the heart of a massive star could be four percentage of its total radius, and most of its mass has contracted and compressed in this tiny ball to capture additional photon particles, increase thermal energy, enhance gravity and central pressure to ignite nuclear fusion processes and burn new elements with releasing huge amount of thermal energy and neutrino particles into space.

The mass of the Sun is very low, and its central temperature could be 15 million kelvins or exceed 80 million degrees when half of its mass is contracted and compressed in its core at 4 percentage of the total radius of a Sun to sustain the nuclear fusion process strongly and form helium or carbon elements. A high mass star has enough mass, density, gravity, pressure, and maximum central temperature to 300 million degrees, which may accelerate nuclear fusion processes for a few million years and produce most of the chemical elements in its heart before its explosion. Most massive stars may explode in a supernova event and feed the universe with crucial chemical materials for life. The remnant core of a collapsed star may become a white dwarf, black dwarf, neutron star, or stellar mass black hole, according to the mass of the dead star. The black hole may grow step by step as it collects enough mass from its surroundings, or its heart is bombarded by superparticles [46–50]. The mass of a black hole increased to become a supermassive black hole and a supergiant black hole due to collecting enough mass. An accretion disc of a black hole heats up steeply due to singularity tunnel waves and superparticles that are propagating through it. The surface temperature of a massive star or a blue star may reach 30 thousand kelvins due to its huge central temperature. Most of the central temperature of stars may be released as coronal mass ejections, stellar flares, powerful plasma jets, neutrino particles escaping, and stellar lightnings. All these dynamical phenomena are involved in coronal heating dynamics and stellar energy evacuation. The surface temperature of a star relies on the mass, central temperature, pressure, gravity, and burning ratio of the chemical elements in the core of a star. The massive stars have higher central pressure and temperature than pressure and temperature of low-mass stars; as a result, their surface temperature is higher than that of low-mass stars. The radius, size, density, pressure, and temperature of stars are changeable due to acceleration and deceleration of the nuclear fusion processes in the heart of Stars and stellar activities. The solar activity phenomena has been increased and repeated every 11 years due to an increments in the ratio of burning hydrogen fuel in Sun’s heart, that is releasing as energetic particles and coronal mass ejection into space. Number of sunspots, solar flares, spicules, plasma jets, solar lightnings, solar wind, and coronal mass ejections had been increased due to solar activity event. Following an equation that make comparisons between the luminosity, radius, and surface temperature of stars with different masses and the Sun [51–57]. According to mass—luminosity relation, the luminosity for massive main-Sequence stars with masses higher than the Sun $(\mathit{M}>\mathbf{M}\mathbf{☉})$ is determined by Equation (9):

${\displaystyle \frac{\mathit{L}}{\mathbf{L}\mathbf{☉}}}={\left(\mathit{ }{\displaystyle \frac{\mathit{M}}{\mathbf{M}\mathbf{☉}}}\mathit{ }\right)}^{3.5}$

           (9)

Indeed, Equation (10) is a necessary equation to determine the luminosity for main-Sequence stars with masses lower than the Sun, typically below 0.43 solar mass, since $(\mathit{M}<\mathbf{M}\mathbf{☉})$:

${\displaystyle \frac{\mathit{L}}{\mathbf{L}\mathbf{☉}}}={\left(\mathit{ }{\displaystyle \frac{\mathit{M}}{\mathbf{M}\mathbf{☉}}}\mathit{ }\right)}^{2.3}$                   (10)

The radius ratio for the star with a mass lower or higher than the mass of a Sun is a crucial relation to determine the radius of a Star directly by Equation (11) according to main composition of a star:

${\displaystyle \frac{\mathit{R}}{\mathbf{R}\mathbf{☉}}}={\left(\mathit{ }{\displaystyle \frac{\mathit{M}}{\mathbf{M}\mathbf{☉}}}\mathit{ }\right)}^{\mathit{a}=(0.6,\mathit{ }0.7\mathit{ }\mathit{o}\mathit{r}\mathit{ }0.8)}$                   (11)

where L is the luminosity of a star as compared to the luminosity of a Sun $\mathrm{L}\mathrm{☉}$. M is the mass of a star, and $( \mathrm{M}\mathrm{☉}=1.99 x{10}^{30} kg)$ is the mass of a Sun.

${\displaystyle \frac{T}{\mathrm{T}\mathrm{☉}}}={( {\displaystyle \frac{L}{\mathrm{L}\mathrm{☉}}} )}^{1/4} x {( {\displaystyle \frac{R}{\mathrm{R}\mathrm{☉}}} )}^{-1/2}$                 (12)

$L=4\pi R^2\sigma T^4$ (13)

${\displaystyle \frac{L}{\mathrm{L}\mathrm{☉}}}={\left( {\displaystyle \frac{R}{\mathrm{R}\mathrm{☉}}} \right)}^2 x {\left( {\displaystyle \frac{T}{\mathrm{T}\mathrm{☉}}} \right)}^4$                 (14)

${\displaystyle \frac{\mathit{T}}{\mathbf{T}\mathbf{☉}}}={\left(\mathit{ }{\displaystyle \frac{(\mathit{L}/(\mathbf{L}\mathbf{☉})}{{\left(\mathit{ }\mathit{R}/\mathbf{R}\mathbf{☉}\mathit{ }\right)}^2}}\mathit{ }\right)}^{1/4}$   (15)

where, σ is the Stefan–Boltzmann constant with a value $(\sigma =5.67 x {10}^{-8}{\displaystyle \frac{W}{m^2.{ K}^4}} ) .$ Then, T is an effective surface temperature of the Star as compared to a Sun’s surface temperature $( \mathrm{T}\mathrm{☉}=5772\mathrm{ }\mathrm{K}\mathrm{ })$. The surface temperature and luminosity of massive stars are incredibly high as compared to the Sun’s Luminosity and its surface temperature.

Problem 4: The massive star with a mass of 20 times the mass of a Sun $(3.98 x {10}^{31} kg)$, and its radius 8 times the radius of a Sun $( 5.57 x {10}^9 m)$, the central pressure in the heart of this massive star is $({\mathit{P}}_{\mathit{c}}=1.28\mathit{ }\mathit{x}\mathit{ }{10}^{18}\mathit{ }\mathit{P}\mathit{a}\mathit{ })$, its central temperature is ${(\mathbf{ }\mathbf{T}}_{\mathbf{c}}=220382469\mathit{ }\mathit{K}\mathit{ }),$ and the surface temperature of a Sun is ( 5772 K) calculate its surface temperature of this massive star T ?.

Solution: Massive main-sequence star with a mass 20 times the Sun has a huge luminosity$\mathit{ }(\mathit{ }{\displaystyle \frac{\mathit{L}}{\mathrm{L}\mathrm{☉}}}={\left(\mathit{ }{\displaystyle \frac{\mathit{M}}{\mathbf{M}\mathbf{☉}}}\mathit{ }\right)}^{3.5}={\left(\mathit{ }{\displaystyle \frac{20\mathit{ }\mathit{x}\mathit{ }\mathbf{M}\mathbf{☉}}{\mathbf{M}\mathbf{☉}}}\mathit{ }\right)}^{3.5}=\mathit{ }{\left(\mathit{ }20\mathit{ }\right)}^{3.5}=3.58\mathit{ }\mathit{x}\mathit{ }{10}^4\mathit{ }).$ Resubstitutethis valuein:

${\left(\mathit{ }{\displaystyle \frac{\mathit{L}}{\mathbf{L}\mathbf{☉}}}\mathit{ }\right)}^{1/4}={\left(3.58\mathit{ }\mathit{x}\mathit{ }{10}^4\mathit{ }\right)}^{1/4}=13.75,$${\left(\mathit{ }{\displaystyle \frac{\mathit{R}}{\mathbf{R}\mathbf{☉}}}\mathit{ }\right)}^{1/2}={\left(\mathit{ }{\displaystyle \frac{8\mathit{ }\mathit{x}\mathit{ }\mathbf{R}\mathbf{☉}}{\mathbf{R}\mathbf{☉}}}\mathit{ }\right)}^{1/2}={\left(\mathit{ }8\mathit{ }\right)}^{1/2}=2.83,$By using Equation (12) the Stefan-Boltzmann law to calculate an effective surface temperature of a massive star T in first method. Substitute above values in this equation to determine an effective surface temperature of a massive star:

${\displaystyle \frac{\mathit{T}}{\mathrm{T}\mathrm{☉}}}={\left(\mathit{ }{\displaystyle \frac{\mathit{L}}{\mathbf{L}\mathbf{☉}}}\mathit{ }\right)}^{1/4}\mathit{ }\mathit{x}\mathit{ }{\left(\mathit{ }{\displaystyle \frac{\mathit{R}}{\mathbf{R}\mathbf{☉}}}\mathit{ }\right)}^{-1/2}.$${\displaystyle \frac{\mathit{T}}{5772\mathrm{ }\mathrm{K}}}={\displaystyle \frac{13.75}{2.83}}$Then surface temperature of a massive star is incredibly high $\left( \mathit{T}=28044\mathit{ }\mathit{K}\mathit{ }\right).$ This star is a hot O-type star or an early B-type star. It has a blue-white appearance, extremely high radiation pressure, and emits a strongly intense electromagnetic spectrum, including infrared, visible light, ultraviolet UV, X-rays, and gamma rays. $({\mathit{\lambda }}_{\mathit{M}\mathit{a}\mathit{x}}={\displaystyle \frac{0.0029\mathit{ }\mathit{m}.\mathit{ }\mathit{K}}{\mathit{T}}}={\displaystyle \frac{0.0029\mathit{ }\mathit{m}.\mathit{K}}{28044\mathit{ }\mathit{K}}}).$ Then maximum wavelength of an emitting radiation from the external surface of a massive star is determined as:$\mathit{ }{\mathit{\lambda }}_{\mathit{M}\mathit{a}\mathit{x}}=1.034\mathit{ }\mathit{x}\mathit{ }{10}^{-7}\mathit{ }\mathit{m}.$By using Equation (14) to calculate surface temperature of massive star in a second method:

${\displaystyle \frac{\mathit{L}}{\mathrm{L}\mathrm{☉}}}={\left(\mathit{ }{\displaystyle \frac{\mathit{R}}{\mathbf{R}\mathbf{☉}}}\mathit{ }\right)}^2\mathit{ }\mathit{x}\mathit{ }{\left(\mathit{ }{\displaystyle \frac{\mathit{T}}{\mathbf{T}\mathbf{☉}}}\mathit{ }\right)}^4$

$3.58\mathit{ }\mathit{x}\mathit{ }{10}^4={\left(\mathit{ }8\mathit{ }\right)}^2\mathit{ }\mathit{x}\mathit{ }{\left(\mathit{ }{\displaystyle \frac{\mathit{T}}{\mathbf{T}\mathbf{☉}}}\mathit{ }\right)}^4$${\left(\mathit{ }{\displaystyle \frac{\mathit{T}}{\mathbf{T}\mathbf{☉}}}\mathit{ }\right)}^4={\displaystyle \frac{35778}{64}}=559.$Take fourth root:

${\displaystyle \frac{\mathit{T}}{\mathbf{T}\mathbf{☉}}}={\left(559\right)}^{1/4}=4.86.$$\mathit{T}=4.86\mathit{ }\mathit{x}\mathit{ }\mathbf{T}\mathbf{☉}=4.86\mathbf{ }\mathbf{x}\mathbf{ }5772\mathbf{ }\mathbf{K}=28052\mathbf{ }\mathbf{K}.$

By using Equation (15) to calculate the surface temperature of a massive star in Third method directly:

${\displaystyle \frac{\mathit{T}}{\mathrm{T}\mathrm{☉}}}={\left(\mathit{ }{\displaystyle \frac{(\mathit{L}/(\mathrm{L}\mathrm{☉})}{{\left(\mathit{ }\mathit{R}/\mathbf{R}\mathbf{☉}\mathit{ }\right)}^2}}\mathit{ }\right)}^{1/4}={\left(\mathit{ }{\displaystyle \frac{\left(3.58\mathit{ }\mathit{x}\mathit{ }{10}^4\right)}{{\left(\mathit{ }8\mathit{ }\right)}^2}}\mathit{ }\right)}^{1/4}\mathit{ }=4.86.$$\mathit{T}=4.86\mathit{ }\mathit{x}\mathit{ }\mathbf{T}\mathbf{☉}=4.86\mathbf{ }\mathbf{x}\mathbf{ }5772\mathbf{ }\mathbf{K}=28052\mathbf{ }\mathbf{K}.$Then surface temperature of a massive star is $\left( \mathit{T}=28052\mathit{ }\mathit{K}\mathit{ }\right).$ It is the same result since determined in Equation (12). The nuclear fusion causes stars to shine. However, any increase has been occurred in a surface temperature and luminosity of stars depends on the stage of the star’s life. Main Sequence (Most of a star’s life), where Fusion of hydrogen into helium in the core provides a huge amount of energy. Fusion in massive stars is violent, producing heavy elements and releasing enough energy into space. The core is incredibly hot, and the surface temperature and luminosity are generally stable. When a star burns through its hydrogen, the helium core begins to contract under gravity. This contraction increases the core temperature, which causes the outer layers to expand drastically. As the star expands, its surface luminosity increases, turning it into a Red Giant. When the star becomes a White Dwarf, its core fusion stops, and it slowly cools, meaning the luminosity decreases due to radius contraction. The surface temperature and luminosity increase most dramatically when a star leaves the main sequence and enters its red giant phase. The white dwarf stars and neutron stars have a smaller radius, but their surface temperature incredibly high, but the luminosity of such compacted objects still lower and dim. The central temperature of the star is higher than its external surface temperature because the central sphere has a smaller radius than the total radius of a star. The fusion ball is only four percent of the total radius of the star where nuclear fusion active and sustainable in the core of a star to enhance its central temperature.

Magnetic Pressure $\left(P_B\right)$: This is the “pressure” exerted by the magnetic field B. It is associated with the magnetic energy density. In a corona, the magnetic field is often structured into loops or filaments.

${\mathit{P}}_{\mathit{B}}={\displaystyle \frac{{\mathit{B}}^2}{{\mathit{\mu }}_0}}$ (16)

The magnetic permeability of free space ${\mu }_0$, also known as the magnetic constant, is a physical constant defined as: $( {\mu }_0=4\pi x {10}^{-7}{\displaystyle \frac{H}{m}}).$ This constant represents the ability of a vacuum to allow magnetic flux to pass through it, acting as the reference for all other materials.

Plasma Thermal Pressure $\left({\mathit{P}}_{\mathit{t}\mathit{h}}\right)$: This is the standard gas pressure of the plasma, where $\mathit{n}$ is the number density of energetic particles and ions,$\mathit{ }{\mathit{K}}_{\mathit{B}}$ is Boltzmann’s constant, and is$\mathit{ }\mathit{T}$ temperature of particles and ions.

${\mathit{P}}_{\mathit{t}\mathit{h}}=\mathit{n}\mathit{ }{\mathit{K}}_{\mathit{B}}\mathit{ }\mathit{T}$ (17)

In the Magnetic pressure and thermal pressure Equilibrium $({\mathit{P}}_{\mathit{B}}={\mathit{P}}_{\mathit{t}\mathit{h}})$ the magnetic field acts like a bottle or container. It compresses the plasma from the sides (transverse to field lines) and provides tension along the field lines. This means that in the solar corona and stellar corona, the magnetic field is strong enough to constrain, heat, and shape the incredibly hot plasma approximately millions of kelvins, preventing it from expanding freely into space. Magnetic field can funnel thermal ionised plasma particles to longer and thinner stellar lightnings or intense, hottest plasma jets to heat up the stellar corona or solar corona. This equation represents the magnetohydrostatic equilibrium (MHS) in stellar coronae, meaning it describes how the magnetic field structure supports and compresses the hot plasma (ionized gas) at a steady state. Magnetic Energy Conversion: The heating of the corona is driven by this magnetic field, often through magnetic reconnection (where magnetic field lines snap and reconnect, releasing energy) or wave dissipation (energy traveling along field lines). Self-Regulating Loop Structures: Magnetic loops are continuously filled with hot plasma. The pressure equilibrium defines the structure of these loops, dictating that regions with stronger magnetic fields generally support higher number density $\mathit{n}$ and temperature T plasmas.

$\mathit{\beta }={\displaystyle \frac{{\mathit{P}}_{\mathit{t}\mathit{h}}}{{\mathit{P}}_{\mathit{B}}}}$

 (18)

If beta lower than 1, $(\mathit{\beta }<1)$, meaning the magnetic pressure $\left({\mathit{P}}_{\mathit{B}}\right)$ is much higher than the thermal pressure $\left({\mathit{P}}_{\mathit{t}\mathit{h}}\right)$, allowing the magnetic field to dominate the coronal heating dynamics. In summary, the formula shows that magnetic energy B dictates the structure and heating of the stellar corona, forcing the plasma into hot, pressurized loops that balance the magnetic pressure, leading to stable coronal structures. This sequence of events describes the behaviour of a magnetically dominated plasma $( \mathrm{l}\mathrm{o}\mathrm{w}-\mathit{\beta }\mathit{ }\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{m}\mathrm{a})$, often seen in astrophysical jets, solar flare filaments, and dense plasma focus experiments. Magnetic Pressure Dominates if $({\mathit{P}}_{\mathit{B}}>{\mathit{P}}_{\mathit{t}\mathit{h}})$: it means the magnetic field strength is high enough to control the plasma dynamics, forcing the plasma to follow field lines and constraining its expansion. Plasma Contracts (Z-pinch/Collimation): because the magnetic pressure exceeds the internal thermal pressure of the plasma, the plasma is squeezed inwards (radial compression or collimation), often forming thin, dense filaments or “ropelike” structures. Filament Brightens: as the plasma is squeezed into a smaller volume, the density increases, which increases the intensity of the emission (increased bremsstrahlung or line emission). Density Spikes Occur: this contraction is not always stable. The magnetic field can pinch the plasma further at certain points, leading to localized, rapid increases in density and temperature, often followed by instabilities that can cause further brightening, heating and luminosity of stars, typically blue stars.

8. Results and Discussion

The mass, radius, luminosity, and surface temperature of stars are changeable over its stellar lifetime, primarily as it moves through different stages of a stellar evolution. While these properties are relatively stable for most of a star’s life, they are fundamentally linked and shift as the star exhausts its nuclear fuel. Red brown stars or red giant stars are cooler, while blue, yellow, and white stars are incredibly hot surface stars. Hotter objects and hotter stars don’t just produce more light and thermal energy; they produce different light or intensify the electromagnetic spectrum. As the temperature increases, the peak intensity of the emitted light or electromagnetic wave shifts toward shorter wavelengths, more energetic wavelengths with higher energy, and higher frequency. The Stefan-Boltzmann Law defines the fundamental relationship between a star’s luminosity, radius, and surface temperature [58,59]. It states that the luminosity is directly proportional to the star’s surface area and the fourth power of its surface temperature. The luminosity depends on the fourth power of temperature, a small increase in temperature results in a massive increase in brightness (e.g., doubling temperature increases luminosity 16-fold). The hotter the surface of a star is, the more light and energy it produces; in physics, this phenomenon is known as thermal radiation (or incandescence). A larger radius means a larger surface area, allowing a star to emit more energy even if its surface temperature is lower than a smaller star. This, combined with Wien’s displacement law, allows astronomers to determine the size of distant stars by measuring their temperature and total energy output. It is important to note that even when an object is not “glowing” red, it is still producing light—it is just producing infrared light (heat radiation) which our eyes cannot see. As it gets hotter, that infrared radiation increases, and eventually, the emission spreads into the visible spectrum. That is exactly right. This phenomenon is known as blackbody radiation, and it explains why temperature and light are so intrinsically linked. Even at room temperature, every object around you—including your own body—is “shining,” just at wavelengths too long for human photoreceptors to detect. This is how thermal imaging cameras work: they act as a different kind of “eye” that can see in the infrared spectrum.

Problem 5: The star with a mass 10 times lower than the mass of a Sun, calculate its surface temperature T ?. Solution: To calculate the surface temperature T, we assume the star is a main-sequence star and use the mass–luminosity and Stefan–Boltzmann relations. Mass of star is $\left(\mathbf{ }\mathbf{M}=0.1\mathbf{ }\mathbf{M}\mathbf{☉}\right).$ Radius of a star is $\left(\mathbf{ }\mathbf{R}=0.16\mathbf{ }\mathbf{R}\mathbf{☉}\right)$ according to Equation (11). The solar surface temperature is $\left(\mathbf{ }\mathbf{T}\mathbf{☉}=5772\mathbf{ }\mathbf{K}\right).$ By using Equation (11) to determine the radius of a low mass star that lower than a Sun:

${\displaystyle \frac{\mathit{R}}{\mathrm{R}\mathrm{☉}}}={\left(\mathit{ }{\displaystyle \frac{\mathit{M}}{\mathbf{M}\mathbf{☉}}}\mathit{ }\right)}^{0.8}$=${\left(\mathit{ }0.1\right)}^{0.8}=0.16.$$\left(\mathbf{ }\mathbf{R}=0.16\mathbf{ }\mathbf{R}\mathbf{☉}\right).$The mass–luminosity for low-mass stars below 0.43 solar mass, $(\mathit{M}<\mathbf{M}\mathbf{☉})$:

${\displaystyle \frac{\mathit{L}}{\mathrm{L}\mathrm{☉}}}={\left(\mathit{ }{\displaystyle \frac{\mathit{M}}{\mathbf{M}\mathbf{☉}}}\mathit{ }\right)}^{2.3}={\left(\mathit{ }0.1\mathit{ }\right)}^{2.3}=0.00501$.

${\left(\mathit{ }{\displaystyle \frac{\mathit{R}}{\mathbf{R}\mathbf{☉}}}\mathit{ }\right)}^2={\left(\mathit{ }{\displaystyle \frac{0.16\mathbf{ }\mathbf{R}\mathbf{☉}}{\mathbf{R}\mathbf{☉}}}\mathit{ }\right)}^2={\left(\mathit{ }0.16\mathit{ }\right)}^2=0.0256.$By using Equation (13) to calculate the surface temperature of a star with a mass lower than the Sun: Substitute values in this equation

${\displaystyle \frac{\mathit{L}}{\mathrm{L}\mathrm{☉}}}={\left(\mathit{ }{\displaystyle \frac{\mathit{R}}{\mathbf{R}\mathbf{☉}}}\mathit{ }\right)}^2\mathit{ }\mathit{x}\mathit{ }{\left(\mathit{ }{\displaystyle \frac{\mathit{T}}{\mathbf{T}\mathbf{☉}}}\mathit{ }\right)}^4$$0.00501={\left(\mathit{ }0.16\mathit{ }\right)}^2\mathit{ }\mathit{x}\mathit{ }{\left(\mathit{ }{\displaystyle \frac{\mathit{T}}{\mathbf{T}\mathbf{☉}}}\mathit{ }\right)}^4$${\left(\mathit{ }{\displaystyle \frac{\mathit{T}}{\mathbf{T}\mathbf{☉}}}\mathit{ }\right)}^4={\displaystyle \frac{0.00501}{0.0256}}=0.1957.$Take fourth root:

${\displaystyle \frac{\mathit{T}}{\mathbf{T}\mathbf{☉}}}={\left(0.1957\right)}^{1/4}=0.665.$$\mathit{T}=0.665\mathit{ }\mathit{x}\mathit{ }\mathbf{T}\mathbf{☉}=0.665\mathbf{ }\mathbf{x}\mathbf{ }5772\mathbf{ }\mathbf{K}=3838\mathbf{ }\mathbf{K}.$The star with mass 0.1 solar mass is a red dwarf star or brown star, cool, dim, spectral type-M, surface temperature about (3000 K to 4000 K), radius 10% of the radius of a Sun. Most of the low-mass stars fail to ignite or start in nuclear fusion processes in their cores due to lower pressure, density, and central temperature. The mass of the star must reach the mass of a sun or over it, and its central temperature must exceed 10 million kelvins to initiate nuclear fusion processes and form new heavy chemical elements. The Sun is an average main-sequence star, and nuclear fusion initiated at its core more than five billion years ago. Then, two hydrogen isotopes fuse in the heart of the Sun, and a new helium atom is formed there in any fusion process. If the star has the same mass as the Sun and it’s assumed to be similar in structure and radius, it would have the same surface temperature. So, the surface temperature would be 5772 K, like the Sun.

Problem 6: The star with a mass 3 times higher than the mass of a Sun, the surface temperature of a Sun is 5772 K, calculate its surface temperature T ?.

Solution: by using Equation (11) the mass—radius relation to determine the radius of a high mass star with a mass 3 times higher than the mass of a Sun$\left(\mathbf{ }\mathbf{M}=3\mathbf{ }\mathbf{M}\mathbf{☉}\right)$:

$( {\displaystyle \frac{M}{\mathrm{M}\mathrm{☉}}}=3),$${\displaystyle \frac{\mathit{R}}{\mathrm{R}\mathrm{☉}}}={\left(\mathit{ }{\displaystyle \frac{\mathit{M}}{\mathbf{M}\mathbf{☉}}}\mathit{ }\right)}^{0.8}$=${\left(\mathit{ }3\right)}^{0.8}=2.41.$$\left({\displaystyle \frac{\mathbf{R}}{\mathbf{R}\mathbf{☉}}}=2.41\mathbf{ }\right).$The mass–luminosity relation for high-mass stars with a mass three times higher than a solar mass, $(\mathit{M}>\mathbf{M}\mathbf{☉})$:

${\displaystyle \frac{\mathit{L}}{\mathrm{L}\mathrm{☉}}}={\left(\mathit{ }{\displaystyle \frac{\mathit{M}}{\mathbf{M}\mathbf{☉}}}\mathit{ }\right)}^{3.5}={\left(\mathit{ }3\mathit{ }\right)}^{3.5}=46.8,$$(\mathit{ }{\displaystyle \frac{\mathit{L}}{\mathrm{L}\mathrm{☉}}}=46.8).$By using Equation (15) to calculate the surface temperature of a massive star directly:

${\displaystyle \frac{\mathit{T}}{\mathrm{T}\mathrm{☉}}}={\left(\mathit{ }{\displaystyle \frac{(\mathit{L}/(\mathrm{L}\mathrm{☉})}{{\left(\mathit{ }\mathit{R}/\mathbf{R}\mathbf{☉}\mathit{ }\right)}^2}}\mathit{ }\right)}^{1/4}={\left(\mathit{ }{\displaystyle \frac{\left(46.8\right)}{{\left(\mathit{ }2.41\mathit{ }\right)}^2}}\mathit{ }\right)}^{1/4}\mathit{ }=1.684.$$\mathit{T}=1.684\mathit{ }\mathit{x}\mathit{ }\mathbf{T}\mathbf{☉}=1.684\mathbf{ }\mathbf{x}\mathbf{ }5772\mathbf{ }\mathbf{K}=9720\mathbf{ }\mathbf{K}.$The main-sequence star with a mass 3 solar mass is typically: spectral type late B and early A, much brighter than the Sun, hotter, its surface temperature about 9724 K. It is a bluer, and shorter life time star. The star with a mass higher than the mass of the Sun may burn its nuclear fuel quickly, and its core may collapse to form a neutron star or black hole. In a seven problem, the star with a mass 10 times higher than the mass of a Sun, calculate its surface temperature T ?. By using Equation (11) the mass—radius relation to determine the radius of a high mass star with a mass 10 times higher than the mass of a Sun $\left(\mathbf{ }\mathbf{M}=10\mathbf{ }\mathbf{M}\mathbf{☉}\right)$: if $( {\displaystyle \frac{M}{\mathrm{M}\mathrm{☉}}}=10),$ ${\displaystyle \frac{\mathit{R}}{\mathrm{R}\mathrm{☉}}}={\left(\mathit{ }{\displaystyle \frac{\mathit{M}}{\mathbf{M}\mathbf{☉}}}\mathit{ }\right)}^{0.7}$=${\left(\mathit{ }10\right)}^{0.7}=5.01.$ Then$\left({\displaystyle \frac{\mathbf{R}}{\mathbf{R}\mathbf{☉}}}=5.01\mathbf{ }\right).$ The mass–luminosity relation for high-mass stars with a mass ten times higher than a solar mass, $(\mathit{M}>\mathbf{M}\mathbf{☉})$: ${\displaystyle \frac{\mathit{L}}{\mathrm{L}\mathrm{☉}}}={\left(\mathit{ }{\displaystyle \frac{\mathit{M}}{\mathbf{M}\mathbf{☉}}}\mathit{ }\right)}^{3.5}={\left(\mathit{ }10\mathit{ }\right)}^{3.5}=3162.$ $(\mathit{ }{\displaystyle \frac{\mathit{L}}{\mathrm{L}\mathrm{☉}}}=3162).$ Substitute values in Equation (15) to calculate the surface temperature of a massive star: ${\displaystyle \frac{\mathit{T}}{\mathrm{T}\mathrm{☉}}}={\left(\mathit{ }{\displaystyle \frac{(\mathit{L}/(\mathrm{L}\mathrm{☉})}{{\left(\mathit{ }\mathit{R}/\mathbf{R}\mathbf{☉}\mathit{ }\right)}^2}}\mathit{ }\right)}^{1/4}={\left(\mathit{ }{\displaystyle \frac{\left(3162\right)}{{\left(\mathit{ }5.01\mathit{ }\right)}^2}}\mathit{ }\right)}^{1/4}\mathit{ }=3.35.$ Then, the surface temperature of this star $\mathit{T}=3.35\mathit{ }\mathit{x}\mathit{ }\mathbf{T}\mathbf{☉}=3.35\mathbf{ }\mathbf{x}\mathbf{ }5772\mathbf{ }\mathbf{K}=19336\mathbf{ }\mathbf{K}.$ Indeed, a 10 solar mass star is typically a hot and luminous blue B-type star. Its surface temperature more than three times higher than a Sun. Real Stellar models usually give 17000 K to 25000 K, so our scaling estimate is reasonable. Furthermore, the star with a mass 25 times the mass of a Sun has a surface temperature about $( T=31000\mathit{ }\mathit{K}).$ The star with a mass 50 times the sun has hottest surface temperature about $( T=45000\mathit{ }\mathit{K}).$Astronomers have identified stars with masses around or even exceeding 200 times the mass of a Sun. The most famous, and currently considered the most massive star known, is R136a1. The most massive star recently known to date, R136a1, with a mass at birth 320 times higher than the mass of our sun [60,61]. Here are the key facts about it. It is located in the Large Magellanic Cloud (a dwarf galaxy orbiting our Milky Way) inside the Tarantula Nebula. They are rich in gas and young stars, featuring active star-forming regions like the LMC’s Tarantula Nebula. Recent, high-resolution observations (as of 2022-2024) estimate its current mass to be around 150 to 230 solar masses. Initial studies suggested it could have been over 300 solar masses at birth. This type of a star is a Wolf-Rayet star, meaning it is extremely hot, luminous, and is losing mass rapidly through a powerful stellar wind. Wolf-Rayet (WR) stars are rare, massive, and highly evolved, characterized by intense stellar winds with speed over 3000 kilometers per second, and surface temperatures of 20,000 K to over 200,000 K. They are among the most luminous stars known, often thousands to millions of times brighter than the Sun. They are, however, relatively rare, with only about 500-2,000 identified in the Milky Way. They represent the final, short-lived stage before a core-collapse supernova, often appearing as nitrogen-rich (WN) or carbon-rich (WC) types due to the loss of their outer hydrogen envelope [62–65]. These massive stars are burning fuel rapidly, live fast and die young, lasting only a few million years. Their spectra are defined by broad emission lines of ionized helium, nitrogen, or carbon, indicating that the outer layers have been stripped away, exposing the hot, helium-fusing core. It is important to note that size (radius and volume) is different from mass. R136a1 is the heaviest (most massive) star, but it is relatively small (about 30-40 times the radius of the Sun) compared to red supergiant stars like UY Scuti, which are much larger in size but far less massive.

Main properties of a UY Scuti Star are Size: Its radius could be 900 times that of the sun. Brightness: It is a very luminous star, roughly 500,000 times brighter than the Sun. Nature: Its fuel finished as a red hypergiant, it is in the late stages of its life cycle, likely ending in a supernova event within a few million years. Visibility: Despite its size, it is not visible to the naked eye because of its distance and the dust obscuring it. Size Constraints: Due to the difficulty in defining the edge of a star’s atmosphere, its size has been debated, and it may not be the absolute largest, with contenders like WOH G64 or VY Canis Majoris. Mass: While immense in size, UY Scuti is not the most massive star, with only about 10 times the mass of the Sun [66,67]. The star with a mass 230 times the sun has the hottest surface temperature about $( T=100213\mathit{ }\mathit{K}).$ It is hot, blue, and luminous, as the star R136a1 is millions of times brighter than the Sun. Massive stars lose mass rapidly through stellar winds or may collapse in a supernova event after a few million years of burning nuclear fuel to form a stellar mass black hole after ultimate death. Powerful plasma jets (specifically spicules) and magnetic reconnection events, stellar flares, and nanoflare-driven events are primary mechanisms responsible for heating the solar and stellar corona to millions of degrees Kelvin. Inject super-hot ionized gas into the outer atmosphere, maintaining its temperature and driving the solar wind. These processes, observed in high resolution by NASA’s Solar Dynamics Observatory (SDO) and the ESA/NASA Solar Orbiter. The mass-radius relation, mass-luminosity relation, and temperature-radius-luminosity relation are crucial mathematical equations for determining the radius, surface temperature, and luminosity of stars. Coronal heating in massive stars (O and B types) is primarily driven by violent, shock-heated stellar winds rather than the magnetic, loop-dominated mechanism found in solar-type stars. High-mass stars lack convective zones, so their X-ray emission arises from instabilities within the radiatively driven wind, which creates millions-of-degrees plasma in the outer atmosphere. Coronal heating dynamics in low mass stars different from massive stars.

Main properties of the Coronal heating dynamics of the massive stars are: Mechanism: unlike the magnetic “braiding” in cool stars, massive star coronal heating relies on the Line-Deshadowing Instability (LDI). This instability causes fast wind shells to collide with slower ones, generating X-ray-emitting shocks throughout the wind. Line-deshadowing instability (LDI) is an intrinsic, strong instability in the line-driven winds of hot, luminous (OB) stars, causing the outflow to become highly inhomogeneous, clumpy, and structured. Magnetospheres: In cases where massive stars have strong magnetic fields (Magnetic Massive Stars), the wind is trapped in the magnetic equator, forming a Centrifugal Magnetosphere and turbulences due to continuous rotation of stars. Dynamic Features: These trapped regions can form “slingshot prominences, dense, cool clouds of neutral gas (10,000 K) that rotate with the star and are trapped in the hot (2–10 MK) stellar corona. Stellar lightnings: The central temperature of the massive stars exceeded 100 million kelvins, causing the hottest particles to rise from the interior shells of stars and display as lightning trees on their surface. Structure: The resulting corona is not uniformly hot, but rather a dynamic, structured environment with both “centrifugal” and “dynamical” magnetospheres, often undergoing cycles of heating and catastrophic cooling (coronal rain) since cold plasma had been dropped on the surface of stars, the hottest and energetic particles in the stellar coronal maybe evaporated into space which called coronal mass ejections. Evolution: The intense X-ray emission, gamma rays, energetic particles releasing, plasma eruptions, stellar flares, spicules, and wind-driven dynamics are directly linked to the rapid evolution and massive energy output from interior shells of these stars. Coronal heating in massive stars (O, B-type) differs fundamentally from low-mass stars, as they lack deep convection zones to drive a solar-like, magnetic dynamo. Instead, X-ray emission about a million kelvins to 10 million kelvins is driven by shocks within strong, radiation-driven stellar winds [68–71]. Key dynamics involve line-deshadowing instability, resulting in shock-heated plasma, and potential magnetic confinement (Magnetospheres). The lifetime of high-mass stars is shorter than that of low-mass stars, because high-mass stars may burn most of their fuel during a few million years, as low-mass stars require burning their fuel for billions of years. The remnant core of a massive star may become a neutron star or a black hole after ultimate death; the remnant core of a solar mass star may give birth to a white dwarf or a black dwarf after billions or trillions of years of its degeneracy pressure. The maximum number of plasma jets and stellar lightnings may be shown on the surface of massive stars during the stellar fusion moment, and near a supernova event.

Indeed, the core of a massive star is composed primarily of silicon and sulfur after completing the Carbon-Nitrogen-Oxygen (CNO) cycle. CNO is a nuclear fusion process began in massive stars. The core temperature of stars rose violently to produce heavy elements near an iron family. The exact temperature for nuclear fusion processes depends on the mass of stars to create light or heavy elements. The star catastrophically collapses and may explode in an event called a Type II supernova after its fuel is exhausted. Its remnant core may become a neutron star or a stellar mass black hole. Silicon burning appear for an 8 to 25-solar-mass star. The period lasts only a few days to a single day before fuel exhaustion. Silicone burning begins when gravitational contraction raises the star’s core temperature to 2– 4 billion kelvin (2-4 GK), after which silicon and other elements can photodisintegrate, emitting protons or alpha particles. Photodisintegration is a process initiated when high-energy gamma-ray photons break existing Silicon-28 nuclei into alpha particles [72,73]. Alpha Capture is another process that starts after the free alpha particles fuse with the remaining Silicon-28 and other chemical elements to create new, heavier elements:

$\mathbf{S}\mathbf{u}\mathbf{l}\mathbf{f}\mathbf{u}\mathbf{r}\mathbf{ }\mathbf{C}\mathbf{r}\mathbf{e}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}:\mathbf{ }{\mathbf{S}\mathbf{i}}^{28}+{\mathbf{H}\mathbf{e}}^4\to {\mathbf{S}}^{32}$

$\mathbf{A}\mathbf{r}\mathbf{g}\mathbf{o}\mathbf{n}\mathbf{ }\mathbf{C}\mathbf{r}\mathbf{e}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}:\mathbf{ }{\mathbf{S}}^{32}+{\mathbf{H}\mathbf{e}}^4\to {\mathbf{A}\mathbf{r}}^{36}$$\mathbf{C}\mathbf{a}\mathbf{l}\mathbf{c}\mathbf{i}\mathbf{u}\mathbf{m}\mathbf{ }\mathbf{C}\mathbf{r}\mathbf{e}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}:\mathbf{ }{\mathbf{A}\mathbf{r}}^{36}+{\mathbf{H}\mathbf{e}}^4\to {\mathbf{C}\mathbf{a}}^{40}$$\mathbf{T}\mathbf{i}\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{i}\mathbf{u}\mathbf{m}\mathbf{ }\mathbf{C}\mathbf{r}\mathbf{e}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}:\mathbf{ }{\mathbf{C}\mathbf{a}}^{40}+{\mathbf{H}\mathbf{e}}^4\to {\mathbf{T}\mathbf{i}}^{44}$$\mathbf{C}\mathbf{h}\mathbf{r}\mathbf{o}\mathbf{m}\mathbf{i}\mathbf{u}\mathbf{m}\mathbf{ }\mathbf{C}\mathbf{r}\mathbf{e}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}:\mathbf{ }{\mathbf{T}\mathbf{i}}^{44}+{\mathbf{H}\mathbf{e}}^4\to {\mathbf{C}\mathbf{r}}^{48}$$\mathbf{I}\mathbf{r}\mathbf{o}\mathbf{n}\mathbf{ }\mathbf{C}\mathbf{r}\mathbf{e}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}:\mathbf{ }{\mathbf{C}\mathbf{r}}^{48}+{\mathbf{H}\mathbf{e}}^4\to {\mathbf{F}\mathbf{e}}^{52}$$\mathbf{N}\mathbf{i}\mathbf{c}\mathbf{k}\mathbf{e}\mathbf{l}\mathbf{ }\mathbf{P}\mathbf{e}\mathbf{a}\mathbf{k}:\mathbf{ }{\mathbf{F}\mathbf{e}}^{52}+{\mathbf{H}\mathbf{e}}^4\to {\mathbf{N}\mathbf{i}}^{56}$Then, several phenomena occur after Nickel and iron elements are created in the core of a massive star. Nickel Decay: Radioactive Nickel-56 later decays into Cobalt-56, then stable Iron-56. Iron-56 has the highest binding energy, or huge energy barrier per nucleon and cannot support further exothermic fusion. The star develops concentric layers of different burning elements around the iron core called by Onion Shell Structure. Cosmic radiations, positrons, neutrinos, beta decay, electromagnetic spectrum, and gamma radiation appear during nuclear fusion processes in the heart of stars. Structural support fails, triggering a Type II supernova explosion, and the core ended within born a neutron star or black hole, this phenomena named a Core Collapse. The atom with atomic number 25 is Manganese (Mn). It is a hard, brittle, silvery transition metal essential for steel production. Manganese is primarily formed in stars through supernova nucleosynthesis, specifically during the explosions of Type Ia supernovae and, to a lesser extent, core-collapse supernovae. Manganese Formation in Massive Stars (Nucleosynthesis) during Supernova Explosions. Manganese is formed in the intense, high-temperature conditions of supernovae, where rapid thermonuclear reactions fuse lighter elements. Type Ia Supernovae are a major source [74,75]. They occur when white dwarf stars (leftover cores of smaller stars) accumulate too much mass and explode. Core-Collapse Supernovae are Massive stars near the end of their lives that produce manganese, which is dispersed into space during their death. During these explosive events, neutrons are released and combine with iron-group nuclei to create manganese and other elements, which are then ejected into the interstellar medium and incorporated into new stars and planets. Advanced Burning in Massive Stars, typically, in very massive stars (≳ 8 solar masses), late evolutionary stages include: Carbon burning, Neon burning, Oxygen burning, and Silicon burning. During silicon burning, nuclear reactions create elements near the iron peak, including: Iron (Fe), Chromium (Cr), Manganese (Mn), and Nickel (Ni). However, most manganese remains locked inside the star until the next stage. The primary source of manganese is supernova explosions, especially: Type II Supernovae (core-collapse). Shock-driven nuclear reactions produce radioactive isotopes like:

⁵⁵Co → ⁵⁵Fe → ⁵⁵Mn

This sequence represents a common radioactive decay chain in which unstable cobalt-55 decays to stable manganese-55 via iron-55. Decay Mode: Primarily positron emission (76%) and electron capture (24%). Half-life: Approximately 17.53 hours. Process: the proton in the cobalt nucleus converts into a neutron, decreasing the atomic number from 27 (Co) to 26 (Fe) while the mass number stays 55.In the massive star explosion ejects manganese into interstellar space. The atom with the symbol Co is Cobalt. It is a chemical element with the atomic number 27, located in Group 9 and Period 4 of the periodic table. Cobalt is a hard, magnetic, transition metal known for its silvery-gray appearance and its use in high-strength alloys and lithium-ion batteries. Powerful plasma jets and stellar lightnings come out from massive stars during the creation of heavy elements in the core of massive stars, and moments before a supernova event, since the external shell of massive stars expands or stars explode in a violent event called a supernova. A burst of neutrino emission is displayed in Type Ia supernovae since electron captures on free protons and nuclei in the hot, dense matter.

In the extreme environments of black hole accretion discs, the hottest surfaces of stars, lightning belts, and plasma jets, atoms are subjected to physical extremes that radically alter their structure. The combination of strong gravitational fields, intense electromagnetic forces, and high thermal energies causes electron orbits to undergo severe Stark broadening, Zeeman splitting, and relativistic distortion. Powerful Electromagnetic Forces (Zeeman & Stark Effects): In these highly magnetized, high-density plasmas, immense magnetic fields and intense electric microfields disrupt the spherical symmetry of standard atomic orbitals. Electrons are pulled into elongated, asymmetrical trajectories as they align with or are repelled by the dominant magnetic field lines [76–80]. High Thermal Energy: The extreme temperatures provide electrons with massive amounts of kinetic energy. This pushes them into highly excited, loosely bound energy states (such as Rydberg states). Because these outer electrons are so weakly bound to the nucleus, the slightest external magnetic or gravitational perturbation causes their orbits to stretch and deform. Relativistic Frame-Dragging & Gravity: Near a spinning black hole, spacetime itself is twisted (Lense-Thirring effect). This distorts the background geometry in which the atoms exist, inherently stretching and shearing the orbitals. The intense gravitational potential well also stretches the atom’s associated emission wavelengths via extreme gravitational redshift. When a high voltage or laser tears electrons free from neutral gas, it leaves behind a highly conductive highway of positively charged ions and free electrons. Because opposite charges attract and like charges repel, the electric current inherently acts as a force that pulls these particles into incredibly tight, organized chains or filaments along the electrical path.

Table 1. Surface Temperature and luminosity of stars with lower and higher masses.

Stellar Mass$\left(\mathit{M}\right)$	Stellar Radius${\displaystyle \frac{\mathit{R}}{\mathbf{R}\mathbf{☉}}}={\left({\displaystyle \frac{\mathit{M}}{\mathbf{M}\mathbf{☉}}}\right)}^{\mathit{a}}$	Stellar Luminosity${\displaystyle \frac{\mathit{L}}{\mathbf{L}\mathbf{☉}}}={\left({\displaystyle \frac{\mathit{M}}{\mathbf{M}\mathbf{☉}}}\right)}^{\left(3.5\mathit{o}\mathit{r}2.3\right)}$	Surface Temperature of Stars  $\mathit{T}=\mathbf{T}\mathbf{☉}{\left({\displaystyle \frac{(\mathit{L}/(\mathbf{L}\mathbf{☉})}{{\left(\mathit{R}/\mathbf{R}\mathbf{☉}\right)}^2}}\right)}^{1/4}$
${2\mathit{x}10}^{29}\mathit{k}\mathit{g}$	$0.16$	$0.00501$	$3838 K$
${2\mathit{x}10}^{30}\mathit{k}\mathit{g}$	$1$	$1$	$5772 K$
${6\mathit{x}10}^{30}\mathit{k}\mathit{g}$	$2.41$	$46.8$	$9720 K$
${2\mathit{x}10}^{31}\mathit{k}\mathit{g}$	$5.01$	$3162$	$19336\mathrm{K}$
${5\mathit{x}10}^{31}\mathit{k}\mathit{g}$	9.5	$7.81 x {10}^4$	$31000 K$
${1\mathit{x}10}^{32}\mathit{k}\mathit{g}$	15.5	${8.84 x 10}^5$	$45000 K$
${4.6\mathit{x}10}^{32}\mathit{k}\mathit{g}$	45	${1.84 x 10}^8$	$100213 K$

Table 1. Surface Temperature and luminosity of stars with lower and higher masses.

Stellar Mass$\left(\mathit{M}\right)$	Stellar Radius${\displaystyle \frac{\mathit{R}}{\mathbf{R}\mathbf{☉}}}={\left({\displaystyle \frac{\mathit{M}}{\mathbf{M}\mathbf{☉}}}\right)}^{\mathit{a}}$	Stellar Luminosity${\displaystyle \frac{\mathit{L}}{\mathbf{L}\mathbf{☉}}}={\left({\displaystyle \frac{\mathit{M}}{\mathbf{M}\mathbf{☉}}}\right)}^{\left(3.5\mathit{o}\mathit{r}2.3\right)}$	Surface Temperature of Stars  $\mathit{T}=\mathbf{T}\mathbf{☉}{\left({\displaystyle \frac{(\mathit{L}/(\mathbf{L}\mathbf{☉})}{{\left(\mathit{R}/\mathbf{R}\mathbf{☉}\right)}^2}}\right)}^{1/4}$
${2\mathit{x}10}^{29}\mathit{k}\mathit{g}$	$0.16$	$0.00501$	$3838 K$
${2\mathit{x}10}^{30}\mathit{k}\mathit{g}$	$1$	$1$	$5772 K$
${6\mathit{x}10}^{30}\mathit{k}\mathit{g}$	$2.41$	$46.8$	$9720 K$
${2\mathit{x}10}^{31}\mathit{k}\mathit{g}$	$5.01$	$3162$	$19336\mathrm{K}$
${5\mathit{x}10}^{31}\mathit{k}\mathit{g}$	9.5	$7.81 x {10}^4$	$31000 K$
${1\mathit{x}10}^{32}\mathit{k}\mathit{g}$	15.5	${8.84 x 10}^5$	$45000 K$
${4.6\mathit{x}10}^{32}\mathit{k}\mathit{g}$	45	${1.84 x 10}^8$	$100213 K$

9. Conclusions

Indeed, Coronal heating dynamics are primarily driven by the interaction between photosphere convective motions and the solar magnetic field. While the exact mechanisms are still being debated (the “Coronal Heating Problem”), scientific consensus points to two or four main processes: Magnetic Reconnection (Nanoflares or spicules), Wave Heating (Magnetohydrodynamic Waves), Plasma Jets, and the Stellar Lightnings. Stellar wind, plasma jets, and stellar lightnings are emanating from the external surface of the low-mass stars and high-mass stars. Massive stars are initiated in the burning of several types of chemical elements and produce much heavier elements, releasing a massive ratio of energetic particles into space at a second. The central temperature of the low-mass stars is lower than 100 million kelvins, but the central temperature of the massive stars could exceed 100 million kelvins to form heavier chemical elements and release powerful plasma jets and multiple types of stellar lightnings into space. The massive stars have the hottest surface temperature, they are the brightest, and most luminous kind of stars with a blue colour. Indeed, the powerful electromagnetic spectrum and violent stellar wind come from massive stars. The particles in the stellar jets can escape from stars at a speed higher than the escape velocity of a star. Scientists believed the spicules, flares, and Alfven waves are essential to heat the solar corona and stellar corona. Powerful plasma jets and Stellar lightning are involved in the coronal heating dynamic of the Sun and stars directly, and enhanced the turbulences in the photosphere and chromosphere to heat up and glow.

This study is funded by myself.

We appreciate the cooperation with the Kurdistan Space Agency.

I declare that I have no conflict of interests.